A 10-µF capacitor is plugged into a 110 V-rms 60-Hz voltage source, with an ammeter in series. What is the rms value of the current through the capacitor?

To find the rms value of the current through the capacitor, we can follow these steps:

1. Identify the given values:

   • Capacitance, $C = 10 \, \mu\text{F} = 10 \times 10^{-6} \, \text{F}$
   • Voltage, $V_{\text{rms}} = 110 \, \text{V}$
   • Frequency, $f = 60 \, \text{Hz}$

2. Calculate the capacitive reactance, $X_C$: The formula for capacitive reactance is: $X_C = \frac{1}{2 \pi f C}$ Substituting the given values: $X_C = \frac{1}{2 \pi \times 60 \times 10 \times 10^{-6}}$ $X_C = \frac{1}{2 \pi \times 60 \times 10^{-5}}$ $X_C = \frac{1}{2 \pi \times 6 \times 10^{-4}}$ $X_C = \frac{1}{0.00377}$ $X_C \approx 265.26 \, \Omega$

3. Calculate the rms current, $I_{\text{rms}}$: The formula for the rms current through a capacitor is: $I_{\text{rms}} = \frac{V_{\text{rms}}}{X_C}$ Substituting the given values: $I_{\text{rms}} = \frac{110}{265.26}$ $I_{\text{rms}} \approx 0.414 \, \text{A}$

Therefore, the rms value of the current through the capacitor is approximately $0.414 \, \text{A}$.